Proove that

Question

m⁵ - 5m³ + 4m is always divisible by 120, m ∈ N

Answer 1

First factor the expression:

 

m⁵ - 5m³ + 4m = m(m⁴ - 5m² + 4) = m(m² - 4)(m² - 1) = m (m-2)(m+2) (m-1)(m+1)

 

Now reorder the terms:

 

(m-2)(m-1)m(m+1)(m+2)

 

That's actually a string of 5 consecutive numbers.  So the original expression is always equal to the product of 5 consecutive numbers.  What does that mean?

 

In any set of 5 consecutive numbers, 1 of them will be divisible by 5.  So 5 is a factor of the expression.

 

Also, at least one of them will be divisible by 3.  So 3 is a factor of the expression.

 

Also, at least 2 of them will be divisible by 2.  So 2² is a factor of the expression.

 

And finally, at least one of the numbers that are divisible by 2 will also be divisible by 4.  So that adds another 2 into the factors.

 

So finally, we can see that the factors will always include 2³, 3, and 5.  In other words, the number will have to be a multiple of 2³ * 3 * 5 = 120.  So the number will be a multiple of 120, which is the same as saying that it is divisible by 120.